Artículo de Clement

8/18/2019 Artículo de Clement

1/17

JOURNAL OF NUMBER THE ORY40, 359-375 (1992)

The Genus Field of an Algebraic Function Field

ROSARIO CLEMENT

Departamento de Matemtiticas, Universidad de1 Pais Vasco,

Apartado de Correos 644, 48080 Bilbao, Vizcaya, Spain

Communicated by D. Goss

Received October 3, 1990; revised February 11, 1991

INTRODUCTION

In this paper our purpose is to present an analogue to the theory of thegenus field of an algebraic number field into the algebraic function fieldscontext, somehow inspired by papers of Goss, Rosen, and Hayes, andtaking as a starting point, the exposition given by Hasse [6] of the genusfield of a quadratic number field.

Let k = IF& 7’) be the field of rational functions over the finite field F,,1a prime dividing q - 1, and K a cyclic extension of k of degree I, so that

K= k(m) with P(T) a polynomial of F,[T]; let 8, be the integralclosure of F,[T] in K, and cc-that we call the infinite prime of k-theprime divisor ofk corresponding to the prime ideal of the subring F,,[ l/T]of k generated by l/T.

In the classical context in whichF is a quadratic number field, the genusfield of F is the maximal abelian extension of Q contained in the Hilbertclass field of F. The notion of Hilbert class held has no proper analogue foralgebraic function fields; there are however several “partial” analogues (see[S], [12]), depending on which features one focuses attention on. Now,

the Hilbert class field of a quadratic number fieldF is the finite abelianextension of F such that the prime ideals of the ring of integers 8, ofFsplitting there are precisely the principal ideals generated by an element ofpositive norm, that is, whose norm is a square in [w. n our context we lookfor a finite abelian extension ofK, denoted by H’+ ), such that the primeideals of 0, splitting in it, are precisely the principal ideals generated by anelement whose norm (with respect tok) is an Z-power in k, , the completionof k with respect to co. In Section 1 we prove the existence of such anextension W+ ). In Sections 2 and 3 we describe explicitly the greatest

abelian extension I- of k contained in H’+ ), and we compute the numberof “ambiguous classes”; the results exhibited in both sections show a greatsimilarity with the classical results for quadratic fields, and we call r the

3590022-314X/92 163.00

Copyright 0 1992 by Academ ic Press, IncAll rights of reproductmn in any for m resewed.


2/17

360 ROSARIO CLEMENT

genus field of K with respect to X-. Finally in Section 4, making use of thereciprocity law, we give a characterization of the prime ideals of 8, which

split in I7

1

We call infinite primes of K the prime divisors CC,, . CC whose restric-tion to k is the prime divisor co-of course i= 1 if CC ramifies or remainsinert in K, and i = I otherwiwse, that is, if CC splits completely in K-k %and K,, will denote respectively the completions of k and K for the prime

divisors CC and coi, and we put

where NKx, x .. x K,,lk, : K,, x . . . x K,, + k, denotes the norm map.

PROPOSITION 1.1. If U, denotes the unit group of the completion K, of Kat a prime ideal p of d,, the index of K*(A x nIPEMaxOKU,) in the idile

group J, of K is inite.

Proof: We have

U, E J,.p Max OR

On the one hand JK/K*(nj=, Kg, x np U,) N Pit OK, the ideal classgroup of 0, which is, of course, finite. (See [3], [lo]).

On the other hand. we have

(r&z, K:,: A)

= (K* n (n;= 1 K:, x np U,) : K* n (A x HP u,))

J-I:=&,:4(U,: CT’,“) ’

where U, denotes the unit group of ~9~ and


3/17

GENUS FIELD IN FUNCTION FIELDS 361

Since (k*, : k*,‘) = l*, from the homomorphism njEI Kz, --t kz/kz:induced by the norm, we obtain (n$= 1Kz, : A) I I*; considering now the

norm N,,,: K + k, we have

so that (U K :U ‘,’ I) 11, and the proposition follows. 1

Remark. If cc ramifies or remains inert inK, (J, : K*(A x np U,)) =1. Pic 8,1; otherwise, that is, if cc splits in K, (J,: K*(A x np U,)) =(12/( U, : U&+,‘-‘)).Pic 8,(.

Since K*(A x HP U,) is an open subgroup of J, of finite index, thatis, an admissible subgroup of JK, we can define HI+’ to be the classfield of K corresponding to this subgroup of J,. Consequently H’+’ isthe finite abelian extension of K, characterized by Gal(H’+‘/K) NJK/K*(A x & U,). Of course, H’+‘/K is unramitied at every prime idealof eK.

PROPOSITION 1.2. Let pO be a prime ideal of 8,. Then pO splits com-pletely in H(+’ .f and only if it is a principal ideal (/I) with NKIk( fi) E k*,‘.

ProoJ: p0 splits completely inH(+) if and only ifK,*, c K*(A x np U,).Let rcObe a uniformizer of K,,; since UPONK*(A x n, U,) it follows that

K;,sK*(Ax~ ++(I, l,...,zn,, I,...)~K*(dxy U,)-+K*

such that

(fl-‘,p-I,..., fi-‘no,fl-’ ,... )~Axn U,.P

Now K,, x ... x KmzN k, Ok K, thus

N,,(a) = NKBkk,,k,,(a 0 1) = NK,~, x x ~,~,k,k ...) ~1) VCIEK

consequently

(b-‘,fl-I,..., b-‘n,,J-I,... )~Axn U,oP

;=, ;Ekt,K/k m

which proves our assertion. 1

COROLLARY 1.3. If (Q( T)) .S a p rzme ideal in F,[ T] which remains nertin K, the ideal p = tI,(Q(T)) splits completely in H(+‘. 1

PROPOSITION 1.4. H(+) is a Galois extension of k.


4/17


5/17


In the following we write Cl 8,= Ik/P(K+), so that we have then the“reciprocity isomorphism”

Cl 0,~ Gal(H’+‘/K)

given by the Artin symbol [a] -+ ([a], H(+)/K) for [a] ~(218,.

2

If, as previously, K = k(m), we write P(T) = aP,( T)” . . . P,( T)iSwhere Pi(T) are manic irreducible polynomials. We prove in this sectionthe following

THEOREM 2.1. The greatest abelian extension of k contained in H(+ ’ is

I-= IF,,(T,,j’m, . . ,,$‘m).

LEMMA 2.2. Let F/E be a finite extension of function fields of onevariable with finite constant field. Let H be an admissible subgroupof theidele group JF and FH its class field, tf E, is the greatest abelian extensionof E contained in FH, the corresponding subgroup of J, is E*N,,(H).

Proof The proof of this well known result [4] is based on the mainproperties of the reciprocity map. The norm map N,,,: J, + J, is open,hence E*N,,(H) is an open subgroup of J,. On the other hand, from thefact that H/F* is a subgroup of C, of finite index (C, denotes as usually theidble class group of F, that is, C, = Jr/F*) it follows that E*N,,,(H)/E*is a subgroup of finite index of NFIE(CF). Since F/E is a finite extension,NFIE(CF) is also a subgroup of finite index of C,, therefore E*N,,(H) is

an admissible subgroup of J,. In order to prove that its class field isprecisely E,, it is enough to show that such a subgroup is contained in thesubgroup of J, corresponding to any abelian extension E’ of E containedin FH. If H’ denotes the admissible subgroup of J, corresponding to theabelian extension FE’fF, we have E*N,,(H)s E*N,,(H’); now thediagram

J, --+ Gal( FE’IF)

NFIEI IJ, --+ Gal( E’/E)

is conmutative, therefore E*N,,(H) is contained in the subgroup of J,corresponding to E’. i


6/17

364 ROSARIO CLEMENT

According to the above lemma, if f denotes the greatest abelianextension of k contained in H I+) the subgroup of J,: corresponding

to r is k*N,,(A x JJP U,). Our ‘next task is to compute [r: li] =(Jk : k*N,,(A x nP U,)); for this purpose we must look first at thebehaviour of the primes in the extension K/k.

PROPOSITION 2.3. The prime ideals of Fy[ T] +vhich ramlyy in K are the(P,(T)) for i = 1, . s. The irzfinite prime cc of k ramtfies in K if and onl?,if I J deg P(T); it splits completely in K/k if and only if 11 eg P(T) anda E Ff’.

Proof It is straightforward to check that the prime ideals{(Pi(T))}i= ~....,s f lF,[T] ramify in K/k. On the other hand, since f is aunity in F,[T], if Diff(B,/F,[T]) denotes the Different of OK relative to%,[T], we have

Diff(e,/lF,[T]) 2 0,(m)’ ‘.

If Q(T) E Fq[ T] is an irreducible polynomial not dividing P(T) and q is aprime ideal of 8, lying over (Q(T)), it is clear that q $ Diff(B,/lF,[T]),hence (Q(T)) doesn’t ramify in K/k.

Next we recall that k,, denotes the completion of k at the infinite prime,so that k, = F,(( l/T)); if u,, denotes the normalized valuation of K corre-sponding to the infinite prime co,, we have

QP(T)) = b,($?%

= -e,.deg P(T),

where eoc s the ramification degree of cc, in K/k.If l[degP(T), lie,, hence e,=l.If lldeg P(T), we have

k,(m) = k,(JaTd+ ... + ad)

=k,(/a+a,(l/T)+ ... +ad(l/T)d);

a+a,( l/T) + . . . +ad(l/T)d is a unity in k, and (l, q) = 1, therefore theextension k, (m)/k ~ is unramified, thus co doesn’t ramify in K/k. Onthe other hand,

CCsplits completely in K/k o [k ~ (m) : k x ] = 1

oa+a,T+ ... +a,(l/T)dEk*,l

oaE IF:’ (by Hensel’s emma). 1


7/17


PROPOSITION 2.4. With the above notations [r : k] = I”+ ‘.

Proof. For any irreducible polynomial Q(T) E LF,[T], recalling thatU,o, denotes the unit group of the completion k,,, ofk with respect to theprime ideal (Q(T)), we write Viz, = {x E U,o,) uo(x - 1) z 1 } where vo isthe discrete valuation of k,,,. Since we know that the only prime ideals of[F,[T] which ramify inK/k are ((P,(T))),= I,.,,,s and that they do it totallyand tamely, according to well-known properties of the norm map in localfields extensions (see [2] or [ 133) we have

k*N,,(Ax,Up)=k* k*lx u(l) F*d’ p x . . . x u(l) F*d ,a xm (PI) 4 Q 1 (Pa) Yeg s n

(Q)+(P,)

Moreover Jk = k*(k$ x &, CT,,,),hence

x ...x U(PJ ydePl) [F*’ P,X

Here ([F,*: Fz’)=l and (k*, : k*,/)=l.Z; besides UC,,,- Uix,x (F,[T]/(pi(W)* yields W,,, : U&j’, IF$P,) = 1, for all i= 1, . s. Therefore theabove index-which is equal ‘to [r : k]-is 121”/1.m

PROPOSITION 2.5. Every extension of K which is unramified in the primeideals of 8, and a composition of cyclic extensions of k of degree 1 is

contained in H(+‘.

Proof We contend that K*(A x J& U,) c K*N,.,,(JKz) for any exten-sion K’/K which is unramified in the finite primes ofK and a compositionof cyclic extensions ofk of degree 1; that will prove the proposition, by classfield theory.

The properties of the reciprocity map give the commutativity of thediagram

A - J, rec. Gal(K’/K)

I

NK/ i

IJk rec. Gal(K’/k) A EJlZ x . . . x ZjlZ;

641:40/3-x


8/17


9/17


PROPOSITION 3.1. H’(G, K(+I) is a group of order l*/e,f, where e,and , denote, as usually, the ramification and residual degree of co in K/k,

respectively.Proof: By Hilbert’s Theorem 90 the cohomology sequence associated to

the exact sequence

1 -+K’+)+K*+K*/K(+‘+ 1

is

I-+ k* + k* + (K*/K’+‘)G -i H’(G, KC+‘) + 1;

therefore H’(G, K’+‘) N (K*/K’+))G. Clearly a(cr)/a~ KC+), VccE K*, andVo E G, hence (K*/K’+‘)G = K*/K (+ ). On the other hand the canonicalmap

K* --, fi K%,, -, fi K&IAj= I j=l

gives the isomorphism

(surjectivity is ensured by the approximation theorem for valuations). Itremains to compute the order of Hi=, Kz,/A; this follows from the exactsequence

1 _) l-I&l KZ, .c- k2, k:--+A k*’ - 1,02 YI;=, ~,,,dIl;= I Kk,)

where JV is induced by the norm-taking into account that(kz : k*,‘) = 1’and that (by local class field theory)

kzb : Nn;=,Km,,k, (I), K’,)) =emfm. 1

PROPOSITION 3.2. H ‘(G, Up ‘) is a group of order 1*/e, f, .

Proof. Since UK/U L+ ’ is a finite group (in fact (U, : U $ ‘) = 1 or l),the Herbrand quotient h(G, UK/U &+ ) is equal to 1, thus h(G, Up ‘) =h(G, U,); but the Herbrand quotient h(G, U,) is well known (see, forinstance, Artin and Tate Cl]): h(G, U,) = e, f,/i. Therefore, in order to


10/17

368 ROSARIOCLEMENT

prove our assertion it will suffice to compute the order of H’(G, I/ ‘,” ‘).Now, considering the Tate cohomology groups we have

fio(G, u;?), = (U~~‘)G/N,,(u~~“) = lFy*/Fq*‘;

hence, since G is cyclic,

IH*(G, U q+‘)l = @‘(G, U;+))l = IriO(G, UL+‘)l =I,

and the result follows. 1

THEOREM 3.3. If Cl OK= I,/P’,f ’ denotes the ideal classes group of 8,

corresponding to the class ield H ( ) of K and G = Gal(K/k), the order of(Cl 13,)~ is I”.

Proof: It is a well-known fact that H’(G, IK) = { 1); hence, the exactcohomology sequence attached to the exact sequence

1 +(P~+‘)“+Z;+(C18,)“+ H’(G, Pp’)+ 1;

if P, denotes the subgroup of Zg consisting of all the fractional ideals thatare extended of fractional ideals of Fy[ T], obviously P, E (Pg ‘)” and wededuce that

is also exact.If we look now at the exact sequence

l+ uIY+‘-&K’+‘+P(K+‘+ 1

we obtain

1 + IF,*+k* + (PL+‘)G + H’(G, U’,“) + H’(G, K’+‘)

+H’(G, Pk+‘)+ H’(G, Ulrf’)+H’(G, KC+‘)+ . . . . (2)

here we have

H’(G, Uk+‘) 2: fi’(G, Uk+‘) =(uI;“)G

vN,,(U’,t’)=IF,*”H’(G, KC+‘) z &‘(G K’+‘) = k*9

NmW’+ ‘I’


11/17


and the map H2(G, Ur’) + H’(G, K’+‘) may be identified to thecanonical map F,*/[F,*’ + k*/N,,(K’+ ‘), which is injective because

lF*nN,,,(K’+‘)=FY *I Therefore we get from (2) the exact sequence4 .1 ~ (p i+‘)G-+H’(G, U’,“,

Pk

--+H’(G, K’+‘) + H’(G, Pk+‘) -+ 1. (3)

In the sequences (1) and (3) all the abelian groups are finite; comparingtheir orders. it follows at once that

,(cl e )“, = IZ:lPA W’(G P’,“)lK l(f%+ ‘)Glpkl

= IWPkl IH’(G K’+‘)I.IH’(G, Uk+“,l ’

applying now Propositions 3.1 and 3.2, and taking into account thatIZg/P,I = l”, which is easily checked, our assertion is proved. 1

Remark. We can see the analogy between the theorem above and thecorresponding result in the classical quadratic number fields setting: inboth cases the number of “ambiguous” classes is equal to the degree of theextension TjK where r is the genus field ofK with respect of k. We cannote however, that, whereas in the classical context every “ambiguous”class contains an “ambiguous” ideal (that is, an ideal invariant under G),this is not always the case for us, as the following example shows. Let ustake k = IF&T) with q odd and K = k(m) where P(T) is a manicirreducible polynomial; we have then I (Cl OK)“1 = 2. If the degree ofP(T)is two and q = 1 mod 4, it is not difficult to check that the canonical mapZg/P, + (Cl 8,)G is not surjective, hence the class different from theidentity in (Cl 8K)G contains no ambiguous ideal.

To conclude this section we shall determine the subgroup of Cl 8, whichcorresponds via the reciprocity isomorphism Cl 8, E Gal(H’+‘/K), to thegenus field r of K. Let T be a generator of G = Gal(K/k); we denote

PROPOSITION 3.4. There is a canonical isomorphism

Cl O,/(Cl OK)‘- i N Gal(f/K).

ProoJ Let us denote Gal(H’+‘/k) = 9. The genus field r ofK is, bydefinition, the greatest abelian extension of k contained in H’+‘, that is,


12/17

370 ROSARIO CLEMENT

I-= (H(+ ‘)“’ is the fixed subfield of H’+’ with respect to 9’, the derivedsubgroup of 9. To prove our assertion it will suffice to show that the image

Sz of (Cl %,)I-~’ via the reciprocity isomorphism Cl 8,~ Gal(H’+‘/K)coincides with 9’.If ([a], H’+ j/K) E Gal(H’+ ‘/K) denotes the Artin symbol of [al-for

[a] E Cl %,we have

([a]‘-‘, H’+‘/K)=([a]‘, H’+‘/K)([a]-‘, H’f’/K)

=t-‘([a],H’+‘/K)r([a],H’+‘/K)~‘E~’

(here f is an arbitrary extension of r toH’+‘); it follows that

Kc r= (H’+‘)“’ c (H’+‘)R~H(+I,

hence

[(H’+‘)” : K] = (Gal(H’+‘/K) : Q) = (Cl 6, : (Cl %,)I-‘).

We contend that (Cl 6, : (Cl BK)‘+‘)=lS; in fact this is an immediateconsequence of the exactness of the sequence

1 -(Cl 6,) G-cl%,=cl%,---+Cl 6,

(Cl %,)7- l- land Theorem 3.3. Since we have seen in Section 2 that [f : K] = I”, wehave r= (H’+‘)“’ = (Hc+))n; hence J2 = 9’. 1

4

We now give a characterization of the prime ideals of eK which splitcompletely in the genus field r ofK, similar to the classical result obtainedby Hasse [6] in the quadratic number fields context. For that purpose, itis useful to make use of the symbol which describes the behaviour of theprime ideals in Kummer extensions L/k of k, just like the Legendre symboldoes for quadratic extensions F/Q; we recall briefly the properties of sucha symbol, giving explicitly a reciprocity law (Proposition 4.1 below).

Let L=k(m) with Q(T) E IFJT], be a Kummer extension ofdegree I of k, and p a prime ideal of lF,[T], such that Q(T)+ p. Ifk,denotes as always the completion of k with respect to the discrete valuationcorresponding to p, and 6 the topological closure of p ink,, the localfield extension kp( m)/k, is unramilied, therefore the Frobeniusautomorphism of $ is characterized by

( -A ,(mYk,) = ($‘i%%P” mod(8,


13/17


where N fi is the cardinal of the residue field of 6 and ‘$ is the maximalideal of the valuation ring of k,(m). Since the residue fields of $ and

p are isomorphic, we have

($3 kpLymWp~(~) ~ (Q(T))(“p- I),,Jm

mod 9;

clearly N p E 1 mod 1, so that both sides of the congruence are in 5,[T],and hence the congruence is also fulfilled mod p. Moreover the L.H.S. ofthe above congruence is an f-root of unity in IF:; hence

DEFINITION. For any Q(T) E lF4[ T] and any prime ideal p of S,[ r]such that Q(T) $ p, the symbol (Q( r)/p), E lFt denotes the (unique) Z-rootof unity such that

EQ(T))‘“P-1’ mod p.

Therefore we have

consequently p splits completely in the extension k(m)/k if and onlyif (Q(WP), = 1.

This symbol fulfills the obvious properties:

6) vQ(T), WT)E E,CTI, Q(T),NT)$p, we have

($3),(~),=("";"'),,

=l-Q(T)+p+JT]/p)*‘,

(iii) (u/p), =Q(~~--I)“, VaE [FQ*;

besides, it can be extended by multiplicity just as in the case of the classicalJacobi symbol: if a is an ideal of Fq[T] and Q(T) E IF,[ T] is prime to a,let (Q(Wh be


14/17


15/17


if u,(Q)= 1 and u,(R)=O,

(Q&=(RQ),‘= .

Now the proposition follows directly from the product formula. m

Remarks. (1) If Q and R are manic the above theorem gives

(2) Proposition 4.1 may be generalized to polynomials Q,R E (Fq[ T]prime to each other but not necessarily irreducible.

Before giving a description of the prime ideals of 8, which split com-pletely in the genus field I-, let us make a simple remark. If for any manicirreducible divisor Pi(T) of P(T), we denote by P*(T) the polynomial( - 1 d’ Pi(T) where d, = deg(P,( T)), we have

(this is clear since q E 1 mod 1 yields $- 1 E F,,).

THEOREM 4.2. A necessary and sufficient condition for a prime ideal p of8, to split completely in the genus field I- of K, is that for i=1, . sthe prime ideal (P,(T)) of F,[T] splits completely in the extension

k(&it%)lk- here g(T) is the manic generator of Np, ideal norm of p

with respect to the extension K/k-and that 1 divides the absolute degree ofp (which coincides with deg g(T)).

Proof We have

p splits completely inr/K o (p, r/K) =1

* (~3 r/K)1 K(m) = 1 Vi = 1, . s and (P, r/K)1 IFy,(r)= 1

- (NP, k($‘%@i)lk) = 1 Vi = 1, . s and (Np, lFq/(T)/k) = 1

-( >P:(T) 1=NP Iwhere 5 is a generator of Fz.


16/17

374 ROSARIO CLEMENT

If nz is the degree of g(T), so that the cardinal of 0,/p is N p = q”‘, bythe reciprocity law proved above we have

the last equality holds obviously if q is even, that is, K of characteristic 2;if q is odd, the exponent of - 1 is

q”-1--d,+md,+Q+l +q+ ... +qm-‘)+,I

I

,y ~~(rnSrn)IO mod 2.

That is, p splits completely in f iff

(Pi(T), k(J%%)l’)= 1 V’i= 1, . sand rCNyP1”‘= 1;

since the last condition holds iff flm, the theorem is proved. 1

ACKNOWLEDGMENT

I am indebted to Dr. J. M. Souto for his suggestions, valuable comments, and constant

support.

REFERENCES

1. E. ARTIN ANDJ. TA TE, “Class Field Theory,” Benjamin, New York/Amsterdam, 1967.2. J. W. S. CA~~ELS AND A. FR~HLICH (Ed.), “Algeb raic Number Theory,” Proceeding s of a

Conference organized by the L.M.S., A cad emic Press , London , 1967.3. C. CHEVALLEY, “Introduction to the Theory of Algebraic Functions of One Variable.”

A.M.S. Surveys, 1951.

4. Y. FURUTA, The genus field and genus number in algebraic number fields, NagoyaMath. J. 29 (1967), 281-285.

5. L. GOLDSTEIN, On prime discrim inan ts, Nagoya Math. J. 45 (1971), 119-127.6. H. HASSE, Zur Geschlechtertheorie in quad ratischen Zahlkorpern, J. Marh. Sm. Japan 3

(1951), 45-51.


17/17


7. D. R. HAYES, Explicit clas s field theory for rational function fields, Trans. Amer. Math.

Sm. 189 (1974). 77-91.8. D. R. HAYES, Explicit clas s field theory in global function fields, in “Studies in Algebra

and Number Theory,” pp. 173-217, Academ ic Press, New York, 1979.9. M. ISHIDA, The genus fields of algebraic number fields, in “Lecture Notes in Math.,”

Vol. 555, Springer-Verlag. Berlin, 1976.

10. M. ROSEN, S-units and S-class groups in algebraic function fields, J. Algebra 26 (1973)98-108.

11. M. ROSEN, Ambiguou s divisor class es in function tields. J. Number Theory 9 (1977).16G174.

12. M. ROSEN, The Hilbert clas s field in function fields, Exposition. Mafh. 5 (1987), 367-378.13. J. P. SERR E, “Corps Locaux,” Hermann. Paris, 1968 .

Documents

Artículo de Clement